Papers

Fully actuated unmanned aerial vehicles (UAVs) are usually certified through rank conditions on a control-allocation matrix or through free-flight tracking performance. For aerial physical interaction, this certification may be incomplete. During sustained contact, part of the available wrench is consumed by the interaction task, and only the residual wrench remains available for stabilization, disturbance rejection, and maneuvering. This paper introduces a control-theoretic framework for contact-persistent full actuation. A rigid-body model on $\R^{3}\times\SO\left(3\right)$ is combined with a morphology-dependent wrench map that captures fixed-tilt, variable-tilt, coaxial, and overactuated multirotor architectures. We define feasible wrench sets under actuator limits, residual wrench sets under task loading, and residual authority margins that strengthen the usual rank-based notion of full actuation. The main result shows that contact-persistent full actuation is equivalent to interiority of the task wrench in the constrained feasible wrench polytope, and that the residual authority radius is exactly the distance to the polytope boundary. We further introduce a signed residual-margin certificate for infeasible and boundary cases, a slack-maximizing allocation certificate, and a robust implementability condition that can be used as a margin-aware safety filter. Numerical evaluation on an abstract tilted hexarotor shows that full row rank alone does not imply feasible contact operation. Intermediate tilt angles preserve residual authority during pushing, whereas small or excessive tilts fail because of lateral-force deficiency or hover-margin loss.

We give complete conjugacy classes of modular group SL(2,Z). Particularly, the conjugacy classes of hyperbolic elements are decided by the proper equivalence classes of indefinite forms, and we give an example. Finally, we describe the coset representatives of centralizer of S, ST, T and hyperbolic elements of SL(2,Z) by regular continued fraction. In conclusion, most Nichols algebras over modular group are infinite-dimensional except Proposition 4.10.

Given $0<a<1$ and any $C^1$, divergence-free, $(-a)$-homogeneous vector field $u_0$ on the punctured plane, we construct a forward self-similar solution of the two-dimensional incompressible Euler equations on the plane with initial datum $u_0$. No smallness or sign condition is required. The solution is obtained as a vanishing-dissipation limit of hypodissipative self-similar profiles. The key estimate is a uniform vorticity profile bound in the critical Lorentz space $L^{\frac{2}{1+a},\infty}$. The resulting velocity has locally finite energy and converges strongly to $u_0$ locally in $L^2$ as time approaches zero. The vorticity preserves its distribution function and converges strongly to the initial vorticity locally in $L^{\frac{2}{1+a}-}$ as time approaches zero.

We propose a general necessary condition for a spinful fermion chain with SU(2) spin-rotation symmetry to be gapped. Specifically, we prove that the expectation value of a properly defined fermionic twisting operator asymptotically approaches unity in any gapped phase with finite ground-state degeneracy, with finite-size corrections bounded by $\mathscr{O}(1/L)$. Consequently, a non-unity value in the thermodynamic limit provides a sufficient criterion for identifying gapless fermion chains. We confirm this criterion using the $s$-wave Bardeen-Cooper-Schrieffer (BCS) Hamiltonian and determinant quantum Monte Carlo (DQMC) simulations of interacting Hubbard models. We further extend the twisting-operator approach to SU($N$)-symmetric fermionic systems, where the gapped ground-state sector must be SU($N$)-singlet and $\langle \hat{\mathcal{U}}^M\rangle=1+\mathscr{O}(1/L)$ by a fermionic twisting operator $\hat{\mathcal{U}}$ with a suitably chosen integer $M$.

Phase-field models based on the Cahn–Hilliard equation coupled with dynamic boundary conditions provide a thermodynamically consistent framework for moving contact line (MCL) problems. Although physics-informed neural networks (PINNs) offer a mesh-free approach for solving partial differential equations, their direct application to MCL problems remains challenging due to long-time error accumulation, sharp interfacial profiles, localized contact line dynamics, and complex contact angle evolution. In this work, we propose MCL-PINNs, a specialized phase-field neural solver designed for MCL problems with dynamic boundary conditions. The method is built on a discrete-time formulation and incorporates several key techniques, including a multi-network time-marching scheme, a relaxed distribution constraint on the neural network outputs, variable scaling for sharply varying solution features, adaptive loss weighting, adaptive collocation sampling with interface extraction, and, when applicable, symmetry preservation through neural network inputs. These techniques improve the capability of the neural solver in resolving sharp interfacial profiles and contact line motion. The proposed method is validated through three numerical examples involving droplet coalescence, shear-induced droplet deformation, and dynamic wetting in a heterogeneous channel. The numerical results show that MCL-PINNs significantly improve prediction accuracy and robustness compared with standard PINNs formulations, enabling reliable resolution of complex interfacial evolution and moving contact line dynamics.

Nielsen and Soelberg proved that a finite subset $A$ of a torsion-free group with $A\cdot A$ having no unique product satisfies $|A|\ge 8$, and exhibited two groups, here $G_1$ and $G_2$, attaining the bound. Nothing quantitative was known about these extremal configurations. We construct exact, independently verified models of both groups and compute the first quantitative invariants at the global minimum. In $G_1$ no $8$-element symmetric witness lies in the radius-$6$ ball ($933$ elements, certified infeasible), while the Nielsen-Soelberg witness lies in the radius-$7$ ball: the global minimum is spread out. In $G_2$, with its natural eight-generator metric, the witness and its inverse are the only two non-UP $8$-sets in the radius-$1$ ball, and the unique-product staircase takes the value $0$ at $n=8$ but $1$ at $n=9$ – the first known minimizer whose square has exactly one uniquely represented element, so the simultaneous failure of t.u.p. and u.p. seen in the Promislow group is not universal. No $(7,9)$ two-sided witness exists in the searched balls, so the Nielsen-Soelberg profile bound may not be sharp. Finally we treat the universal group $G_3$. Its structure is known – Soelberg's thesis identifies an index-$8$ Heisenberg subgroup of step $8$ and proves torsion-freeness, and Gardam, studying the same group as an amalgam of Klein bottle groups, shows it to be virtually nilpotent but not virtually abelian – and what we add is a model in search coordinates in which balls can be enumerated. In it we reproduce the Nielsen-Soelberg two-sided pair and exhibit a symmetric $15$-element witness whose trivial-coset singleton generates the centre of that Heisenberg subgroup. It is rigid and rare: within $B(5)$ the size $15$ is exactly minimal, the coset profile is forced, and exactly four such witnesses exist in $B(4)$, one orbit. Hence $m_1(G_3)\in[8,15]$ against $m_2(G_3)=16$.

We introduce an infinite family of mean-value formulas (exact and asymptotic) given in terms of linear combinations of iterated means. We prove that the mean-value formulas in this family characterize real-valued polyharmonic functions of finite order, and that a simple algebraic condition partitions the family into equivalence classes according to the order of polyharmonicity. Our key results include strong converses to the mean-value properties – locally integrable functions satisfying a mean-value property in the family are polyharmonic – and a regularity result – locally integrable functions satisfying a mean-value property in the family, whether exact or asymptotic, are smooth.

Break-resilient codes enable reliable communication in the presence of an omniscient adversary that may split a transmitted message at arbitrary boundaries between consecutive symbols, while the receiver observes only an unordered multiset of the resulting fragments. For binary codewords of length~$n$ subject to at most~$t$ breaks, the best known explicit construction has redundancy~$O(t\log_2 n\log_2\log_2\log_2 n)$, whereas the information-theoretic lower bound is~$\Omega(t\log_2(n/t))$. In this paper, we extend the binary break model to any fixed finite field~$\bbF_q$ and establish a redundancy lower bound of~$\Omega(t\log_q(n/t))$. We then give an explicit construction of~$q$-ary break-resilient codes with redundancy~$O(t\log_q n)$ when~$t\leq n^{1-\varepsilon}$ for a fixed constant~$\varepsilon\in(0,1)$, matching the information-theoretic lower bound up to a constant factor. The key idea is to compute a short algebraic fingerprint of the message, which enables the decoder to reject incorrect assemblies of the received fragments.

In multivariate extreme value analysis, the tail dependence between some of the risk variables at hand may be weak, even when other variables do tend to become large simultaneously. Weak tail dependence may induce a substantial bias in estimation procedures based on the limiting multivariate (generalized) Pareto distribution of excesses over high thresholds. We consider a Hüsler–Reiss multivariate generalized Pareto model and, motivated by this issue, propose first- and second-order moment estimators of its variogram matrix constructed from a lower-tail-clipped version of the underlying random vector. The asymptotic normality of the proposed estimators is established. We demonstrate by simulation studies that they have lower bias than the empirical variogram estimator in certain cases, particularly when the dependence between components is weak. The estimators are applied to flood discharge data from the Danube river basin and the US flight delay data, showing that the tail dependence structure implied by the fitted model based on the first-order clipped moment estimator aligns more closely with the empirical tail dependence of the data than that based on the empirical variogram estimator.

We study bounded entire solutions $u:\mathbb{R}^2\to \mathbb{R}^2$ that minimize the Allen-Cahn functional \begin{equation*} J(u,\Omega)=\int_\Omega \left(\frac12 |\nabla u|^2+W(u)\right)\,d\mathbf{x}, \end{equation*} with the $D_3$-invariant triple-well potential \begin{equation*} W(u_1,u_2)=|u|^4+2u_1u_2^2-\frac23 u_1^3-|u|^2+\frac23. \end{equation*} We obtain a complete classification of entire minimizing solutions. In particular, when $u$ has a triple-junction structure at infinity, up to translation and orthogonal change of coordinates, $u$ has the explicit profile \begin{equation*} u_*(\mathbf{x})=\sum_{i=1}^3 \frac{e^{\sqrt2 a_i\cdot \mathbf{x}}}{\sum_{j=1}^3 e^{\sqrt2 a_j\cdot \mathbf{x}}}a_i. \end{equation*} We also demonstrate that the solutions obtained by minimizing within the $D_3$-equivariant class coincide with $u_*$. The key ingredient is a calibration identity arising from the special algebraic structure of $W$.

The class of 2-distance-transitive graphs naturally generalizes distance-transitive graphs and plays a central role in algebraic graph theory. Classifying such graphs for a prescribed underlying group is a key open problem. A vertex-transitive graph $\Gamma$ is said to be $2$-distance-transitive if, for each $i\in \{1,2\}$, any two pairs of vertices with identical distance $i$ in $\Gamma$ can be mapped to each other via some automorphism of the graph. In this paper, we present a complete classification of all $2$-distance-transitive Cayley graphs of the semi-dihedral groups.

In this paper, we study the smallest gaps between successive zeros of nondegenerate smooth stationary centered Gaussian processes on the real line with the assumption that the covariance kernel $\kappa(x)$ and its derivatives decay to 0 as $|x|\to\infty$. We prove that, after rescaling, the smallest gaps converge to a Poisson point process with a specific rate. Moreover, the positions where these smallest gaps occur tend to a uniform distribution. Consequently, we can derive the limiting density for the $k$-th smallest gap.

In this paper, we study the random walk on the symmetric group $\mathfrak{S}_n$ generated by the conjugacy class of $k$-cycles, where $2\le k=o(n/(\log n)^4)$. We prove that the walk exhibits hitting-time mixing: at the first time when every card has been touched, the distribution is already close to equilibrium. For odd $k$, the equilibrium measure is the uniform measure on $\mathfrak{A}_n$. For even $k$, the walk first mixes to the parity mixture determined by the hitting time, and in our range this mixture is asymptotically $U_{\mathfrak{S}_n}$. Our argument combines a refined fixed-time approximation for the random $k$-cycle walk near the cutoff window with an auxiliary marking scheme inspired by Jain-Sawhney's work (arXiv:2410.23944) on random transpositions. The main new feature is a parity-compatible coupling which handles both odd and even $k$-cycles in a unified framework. We also prove a hitting-time mixing result in the opposite regime $k\ge n-o(n^{1/2})$, and formulate a conjecture for all $2\le k\le n-1$.

We characterize traversable wormholes in Unimodular Gravity (UG) and investigate what distinguishes them from their General Relativistic (GR) counterparts. For a class of static and spherically symmetric solutions, we analyze their embedding and timelike geodesics, showing that the geodesic structure is entirely determined by the metric and is therefore identical in both theories. To identify the physical origin of their differences, we compare the source sectors required to sustain the same wormhole geometry. We find that preserving a fixed UG geometry while imposing energy-momentum conservation requires restricting the equation of state, whereas relaxing this condition introduces an effective inhomogeneous vacuum contribution. We further show that the degree of exoticity is preserved even when energy-momentum is not conserved, while departures from the GR sector are encoded in an effective inhomogeneous vacuum structure whose asymptotic behavior resembles that of a cosmological constant. Our results reinforce that UG is geometrically equivalent to GR, while its distinctive features emerge in the dynamical interpretation of the source sector. More generally, our analysis illustrates that different gravitational theories may give rise to identical spacetime geometries while requiring different sources to sustain them.

We study associative smoothness through representation homology with coefficients in arbitrary finite-dimensional algebras. We prove that the higher representation homology of a finitely generated formally smooth algebra with arbitrary finite-dimensional coefficients vanishes. We further show that allowing arbitrary coefficients yields a strictly stronger smoothness test: suitable non-matrix coefficients detect nonsmoothness in situations where matrix coefficients do not. These results raise the question of whether representation homology with arbitrary coefficients furnishes a homotopical characterization of associative smoothness in the spirit of the derived Kontsevich-Rosenberg principle. For finite-dimensional algebras, we prove that representation homology with arbitrary coefficients completely characterizes formal smoothness, providing supporting evidence for this philosophy.

We consider a stochastic opinion dynamics model on a fully connected social network with $N$ actors interacting by expressing opinions from a set of $M$ opinions. At any time $t\geq 0$, each actor is associated to an $M$-tuple representing the social pressure exerted on this actor for each opinion. The evolution of the matrix containing the social pressure of all actors for all opinions is a Markov jump process. Each actor tends to express opinions according to their social pressure vector and this tendency is modulated by a polarization coefficient. When an actor expresses an opinion $o$, its social pressure for all opinions is reset to zero, while for other actors the social pressure for $o$ increases by 1 and the social pressure for other opinions decreases by $1/(M-1)$. In this setting, we prove fast consensus formation, existence of a unique invariant measure and metastability in a highly polarized network. Moreover, by considering a communication bias parameter, the system exhibits a phase transition described as follows. With a negative communication bias parameter, all actors except one stop expressing in a finite time almost surely. Otherwise, no actor stops expressing opinions.

We establish the global well-posedness of the Cauchy problem for the reverse space-time nonlocal Fokas-Lenells equation with the weighted Sobolev initial data $q_0(x)\in H^{3}(\mathbb{R}) \cap H^{2,1}(\mathbb{R})$ on the line. We develop the inverse scattering transform formulated via the associated Riemann-Hilbert problems to study this issue. A spectral uniformization transform is introduced to resolve the singular behavior inherent in the KN-type negative flow spectral problem. Owing to the reverse space-time reduction, reflection coefficients no longer satisfy the usual Hermitian conjugation symmetry, and the coercivity of the jump matrix is therefore not available a priori. The quantitative smallness condition on the initial data yields uniform bounds on the reflection coefficients and ensures the uniform positive definiteness of the Hermitian part of the associated jump matrix. The resulting coercivity allows us to establish the bounded invertibility of the associated singular integral operator through a Fredholm and vanishing-lemma argument. Under this condition, we prove an $L^{2}$-Sobolev bijective correspondence between the potential and scattering data, exclude spectral singularities on continuous spectra, and obtain the global existence and uniqueness of solutions. Moreover, the associated solution map is Lipschitz continuous on the admissible initial-data class.

We establish universal complexity guarantees for quadratic trust-region methods and identify a common mechanism underlying their universal behavior under convexity, based on a function-gap model-decrease estimate that appears to be new in the trust-region literature. First, we prove that the basic trust-region method with inexact subproblem solves is universal under convexity. Under a $\nu$-Hölder-continuous Hessian, it attains the global complexity bound $\mathcal{O}(\varepsilon^{-1/(1+\nu)})$ for computing an $\varepsilon$-approximate minimizer, without knowledge of $\nu\in[0,1]$ or the corresponding Hölder constant. In the nonconvex regime, the method retains the classical $\mathcal{O}(\varepsilon^{-2})$ first-order complexity bound under the usual additional bounded-Hessian assumption. With suitably vanishing inexactness, it also recovers Q-superlinear local convergence for $\nu=0$ and convergence of order $1+\nu$ for $\nu\in(0,1]$. Second, we show that the same convex-universal mechanism applies to a trust-region variant with exact subproblem solves and a simple modification of the acceptance ratio. This variant is universal simultaneously in the nonconvex, convex, and local regimes: it attains the optimal nonconvex first-order complexity $\mathcal{O}(\varepsilon^{-(2+\nu)/(1+\nu)})$, while preserving the universal convex complexity and the local Newton rates. These guarantees require no knowledge of $\nu$ or its Hölder constant. Both methods use the usual quadratic trust-region model and the classical radius-update mechanism, without gradient-dependent radii or model modifications such as cubic, gradient, or tensor regularization. The results show that the trust-region mechanism is inherently adaptive across nonconvex, convex, and locally strongly convex regimes, providing further theoretical support for the practical success of trust-region methods.

This article develops a theory of one-parameter semigroups for linear relations, introducing the notion of domain-multivalued orthogonal (d.m.o.) relations. We establish complete characterizations for generators of uniformly and strongly continuous semigroups, showing them to be maximal and densely d.m.o. relations, respectively. Furthermore, the Hille-Yosida and Lumer-Phillips theorems are extended to this general setting, providing necessary and sufficient conditions for generating $C_0$-semigroups of contractions (identified as maximal anti-accumulative relations) and unitary operators (anti-selfadjoint relations). This rigorous framework sheds new light on the peculiarities of one-parameter semigroups for linear relations.

Motivated by stochastic model predictive control applications, we develop a semialgebraic framework for constraint tightening in chance-constrained systems with unbounded additive disturbances and saturated inputs. The saturation nonlinearity is represented exactly through its semialgebraic description. A polynomial Lyapunov function satisfying a stochastic drift condition is then synthesized using sum-of-squares optimization, yielding finite-time probabilistic reachable sets and probabilistic ultimate bound. The resulting sets are further optimized through an objective tailored to constraint tightening. Numerical examples illustrate the effectiveness and scalability properties of the approach.

Let $f_\alpha(x)=(x^q-(1-\alpha))^d-\alpha^d x^z$, where $0<\alpha<1$, $q\ge 2$, $d\ge 2$, $1\le z\le q-1$, and $\gcd(q,z)=1$. These are the non-degenerate full-degree Type II reduced Ito polynomials of order $n=qd$. We give a complete parametrisation, up to permutation similarity, of the stochastic matrices whose characteristic polynomial is $f_\alpha$. The support data are a composition $a_0+\cdots+a_{d-1}=z$ and a cyclic origin $\rho\in\mathbb{Z}_q$. The transfer support in block 0 is an arbitrary nonempty subset of $[\rho,\rho+a_0]_q$, while the support in block $t\ge 1$ is an arbitrary nonempty subset of $[\rho-\sum_{j=1}^t(a_j+1),\rho-\sum_{j=1}^t(a_j+1)+a_t]_q$. The horizontal weights at the selected positions are arbitrary elements of $(0,1)$ whose product in every block is $1-\alpha$. The proof first shows that the order-$n$ Type II arc lies strictly outside $\Theta_{n-1}$, which permits the Dmitriev–Dynkin/Kirkland–Smigoc two-shift reduction. A circular separation theorem then classifies the possible supports, and Coates' formula supplies the characteristic polynomial. This resolves the non-sparse full-degree Type II characteristic-polynomial realiser problem.

The Roman Domination Problem (RDP) on a graph \(G=(V,E)\) asks for a labeling function \(f:V\rightarrow\{0,1,2\}\) such that every vertex assigned value \(0\) is adjacent to a vertex assigned value \(2\). The objective is to minimize the total weight \(\sum_{v\in V} f(v)\); this minimum value is the Roman domination number of \(G\), denoted by \(\gamma_R(G)\). In this paper, we study RDP on graph classes motivated by convexity and induced-\(P_4\) structure. First, we consider circular-convex bipartite graphs, a natural superclass of convex bipartite graphs, where RDP is already known to be polynomial-time solvable. Assuming that a circular-convex representation is given, we compute \(\gamma_R(G)\) in \(O(n^6)\) time by cutting the circular order, separating interval and wrap-around vertices, and branching over at most two wrap-around vertices assigned value \(2\). Second, we study triad-convex bipartite graphs, a restricted subclass of tree-convex bipartite graphs whose convexity tree is a subdivision of \(K_{1,3}\). Although RDP is hard on broader tree-convex subclasses such as star-convex and comb-convex bipartite graphs, we show that \(\gamma_R(G)\) can be computed in \(O(n^7)\) time on triad-convex bipartite graphs. Finally, we study \(P_4\)-tidy graphs, which properly extend cographs. Using the Giakoumakis et al. structural decomposition of \(P_4\)-tidy graphs, we give a direct, exact algorithm that computes \(\gamma_R(G)\) in \(O(n+m)\) time. These results extend the algorithmic boundary of Roman domination on convexity-based bipartite graphs and \(P_4\)-structured graph classes.

We develop nonparametric inference for reliability indicators of discrete-time semi-Markov systems from independent trajectories observed over a common fixed horizon. Augmenting the physical state by the backward recurrence time yields a finite coupled Markov representation on the observed age range. We distinguish the resulting age-restricted failure-or-exit time from calendar truncation, since these two finite-horizon quantities coincide only in special cases. The framework covers restricted factorial moments and moment characteristics, calendar-truncated failure-time summaries, and the discrete-time intensity of the hitting time. Under explicit row-exposure conditions, strong consistency and joint asymptotic normality are established for the empirical initial law, the required transition rows and the corresponding plug-in functionals. The Gaussian random-matrix representation gives pointwise and joint covariance formulas, simultaneous confidence envelopes, Wald procedures for linear summaries, and curvature-adjusted Gaussian approximations. Restriction diagnostics and a target-specific horizon-selection rule based on exposure, boundary interaction and nested-horizon stability are developed separately. Numerical experiments assess the inferential formulas and the diagnostics, while a complete-case illustration from the European Group for Blood and Marrow Transplantation reports calendar-truncated failure-time summaries and finite-dimensional hitting intensities.

We prove that, for every fixed degree $d$ and a uniformly random simple $d$-regular graph $G_{n,d}$ on an even number $n>d$ of vertices, $$\mathbb{E}\operatorname{MaxCut}(G_{n,d})-\mathbb{E}\operatorname{MaxBis}(G_{n,d})=o(n).$$ In other words, requiring the two sides of a cut to have exactly the same size changes the expected optimal cut by only a sublinear number of edges. We prove this first for the configuration model by comparing cuts of each possible cardinality on an $n$-vertex graph with bisections of a related graph on $2n$ vertices. An important technical tool is Huang's interpolation theorem (Huang, 2018); to apply it, we establish a structural property of the change in the optimal cut when a single edge is added. This comparison, together with concentration estimates and Huang's convergence theorem for maximum bisection, shows that maximum cut and maximum bisection have the same limiting density. Conditioning the configuration model on being simple then gives the result for uniformly random simple regular graphs.

We introduce a capacity-constrained Wasserstein barycenter problem in which each transport plan from an input measure to the barycenter is bounded by a prescribed capacity measure. On compact domains, we prove existence of capacity-constrained barycenters and establish a strong duality formula. We then study an entropy-regularized version under bounded capacity-density assumptions. The regularized problem has a unique optimal tuple of transport plans and a unique barycenter, while its dual involves an explicit capped-exponential penalty. Whenever dual maximizers exist, the optimal densities satisfy a capacity-clipped Gibbs formula. Finally, we prove an $O(\epsilon)$ estimate for the optimal values and show that the regularized plans converge to the minimum-entropy optimal solution of the unregularized problem.

Belyi's theorem asserts that a smooth projective curve over $\mathbb{C}$ is defined over a number field if and only if it admits a non-constant morphism to $\mathbb{P}^1$ ramified over at most three points. This article gives a complete, self-contained account of the theorem – both implications, with all descent machinery proved rather than quoted – together with a study of the objects the proof produces. Beyond the exposition it contains original results, proved in full. The principal one is a structure theorem for the Belyi polynomials $P_{m,n}(z)=\frac{(m+n)^{m+n}}{m^{m}n^{n}}z^{m}(1-z)^{n}$: their dessins are double stars, they satisfy the exact identity $P_{m,n}=\pi_{k}\circ P_{m/k,n/k}$ with $k=\gcd(m,n)$ and $\pi_{k}(w)=w^{k}$, and their monodromy is the wreath product $\mathfrak{S}_{(m+n)/k}\wr\mathbb{Z}/k$, the full symmetric group precisely when $\gcd(m,n)=1$. Around the sharp lower bounds $d\geq 2g+1$ (all Belyi maps) and $d\geq 4g$ (clean maps) we study the extremal maps attaining $d=2g+1$: their monodromy lies in the alternating group, they need not be cyclic – the smallest non-cyclic one has degree $5$, genus $2$, monodromy $A_{5}$, and is defined over $\mathbb{Q}$ – and they satisfy the exact mass formula $\sum 1/|\mathrm{Aut}|=2(d-1)!/d(d+1)$, obtained from Boccara's cycle-factorization count – reproved by a self-contained Frobenius computation – and verified by complete enumeration for $d\leq7$. We also bound the degree of the rationalization step of Belyi's algorithm by $N!$ in the number $N$ of irrational branch values. The theory is illustrated by fully computed examples, and by an explicit $G_{\mathbb{Q}}$-orbit of three plane trees whose fields of moduli are the three conjugate embeddings of the non-Galois cubic field $\mathbb{Q}(\sqrt[3]{2})$.

In this paper, we propose a hybrid-field channel tracking algorithm for extremely large-scale multiple-input multiple-output (XL-MIMO) systems with mobility, leveraging the historical channel state information. The multiple path scenario is considered, and the line-of-sight path is coarsely estimated through the sparse peak search within a narrow window in the fractional Fourier domain based on the temporal continuity of the user's motion. Moreover, coarse estimations of the non-line-of-sight (NLoS) paths are determined by ensuring the scatterer survival probability, eliminating the necessity of detecting existing NLoS paths. Then, a Newton-based refinement is applied to the estimated paths before seeking possible new paths. Numerical results validate that the proposed algorithm achieves superior performance with low computational complexity.

Green's arithmetic regularity lemma over $G=\F_2^n$ has tower-type lower bounds. We show that the large order properties in the Hosseini–Lovett–Moshkovitz–Shapira (HLM) construction survive sparse edits. If $d$ is the top block dimension, then changing at most $|G|/16$ points of the top HLM set $A_s$ still leaves a half-graph of height $d/8$. For a suitable choice of the HLM maps, changing at most $\eps|G|$ points of a superlevel set of the averaged witness still leaves a half-graph of height $d/(8\sqrt{s})\ge{\rm twr}(s-2)$. The proofs use Sauer's lemma and the binary code formed by the top-level row traces.

In short, we call a polyhedron P a height function if there exists some face F of P such that P lies entirely within F when orthogonally projected onto the affine plane containing F. Because tetrahedra come in various configurations, we propose a dihedral angle classification of tetrahedra, and prove that a tetrahedron T is either a height function already, or, at most one planar bisection of T is needed to produce two height function tetrahedra. Furthermore, we prove that such bisections belong to infinitely large families of bisections.

In this paper, we study when the core inverse in a ring coincides with its hybrid (b,c)-inverse, i.e., hybrid core-(b,c)-inverse. We present many characterizations of hybrid core-(b,c)-inverse. To establish a broader framework for this generalized inverses, we define hybrid core-EP-(b,c)-inverse that serves as a natural extension of the hybrid core-(b,c)-inverse. We characterize this new generalized inverse by combining the hybrid core-(b,c)-inverses and quasinilpotents. This generalized inverse is thereby examined through a novel limit-based approach. Its polar-like properties and image-based representations are presented.